Congruences corresponding to hypergeometric identities I. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mmultiscripts><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:none /><mml:mprescripts /><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:none /></mml:mmultiscripts></mml:math> transformations

Mao, Guo-Shuai; Pan, Hao

doi:10.1016/j.jmaa.2021.125527

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Proof Of Theorem 111

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Cited by 5 publications

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“…In view of (2.2), we have where is the gamma function. In view of [7, theorem 14] and [5, (2.4)] (or [10, (3.2)]), for , we have and where denotes the -adic derivative of , denotes the least non-negative residue of modulo , i.e. the integer lying in such that .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

On two congruence conjectures of Z.-W. Sun involving Franel numbers

Mao¹,

Liu²

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$ , then \[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \] and if $p\equiv 1\pmod 3$ , then \[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \] where $f_n=\sum _{k=0}^n\binom {n}k^3$ stands for the $n$ th Franel number.

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